They threatened to open the scoring inside a minute when Marvin Emnes released Stewart Downing and his cross had to be poked behind by Gareth Barry.
Julio Arca headed straight at Brad Friedel before Gary O'Neill's fierce drive deflected off Tuncay and narrowly over the bar.
Villa, who drop into sixth after Everton beat West Ham, were struggling to establish any sort of rhythm yet might have taken the lead.
Gabriel Agbolahor collected Barry's ball over the top but shot straight at the feet of Brad Jones when one-on-one with the Australian goalkeeper.
It was a let-off for the hosts but they continued to push forward and gained their reward.
Southgate hopes for final day drama
Arca fed Downing on the left and his shot looped up off James Milner into the heart of the area, where Tuncay produced a stunning bicycle kick that flew past Friedel's left hand and nestled in the bottom corner.
It was a magnificent effort from the Turkish striker, who grabbed both goals in Boro's 2-1 win at Villa Park earlier in the season.
Southgate's team were knocked out of their stride by the loss of Downing, who hobbled off with a nasty-looking injury following a heavy challenge by Stiliyan Petrov.
Villa then enjoyed a lengthy period of possession but the home defence, boosted by the return of David Wheater, who was playing despite requiring knee surgery, held firm.
That allowed Boro to go in search of a second and, moments after referee Mike Riley rejected a penalty shout as Carew climbed all over Emnes, Friedel made a fine save to deny Justin Hoyte from the edge of the box.
Villa arrived with only one win in their previous 14 games in all competitions but boast one of the division's finest away records - and after half-time they showed exactly why.
O'Neill admits to late season tiredness
The visitors got forward in numbers and applied sustained pressure on a Boro backline that was beginning to creak.
Carlos Cuellar miscued a header and Barry a volley before Boro's failure to clear Petrov's close-range shot allowed Carew to lash his drive past a despairing Jones.
Knowing that anything less than a victory would see them all-but relegated, Boro surged forward but left themselves exposed to the counter-attack.
Petrov screwed a shot wide, Ashley Young was foiled when through on goal, Wheater cleared off the line from an Agbonlahor header and Jones saved well from the same man.
Boro's only significant effort of the second half saw Tuncay glance a header over from Hoyte's left-wing cross.
In Villa they met an organised and stubborn defensive unit and, given the number of openings they created late on, Martin O'Neill's side will feel they should have taken all three points.
Middlesbrough manager Gareth Southgate: "We're relying on everything else going for us and it's obviously a very difficult position for us.
"We've got to keep going. If we don't then we've got no chance. If it doesn't go for us then so be it.
"On the last Sunday of the season there can be many twists and turns. We've got to prepare and try to beat West Ham."
Aston Villa manager Martin O'Neill: "We had chances to finish the game but Middlesbrough were pressing so there was always that possibility.
"We were a bit lethargic in the first half but in the second we got our hands dirty and we performed very well.
"I would admit to a bit of tiredness - it's been a long season but we'll try and finish as strongly as we can."
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